Ann. Funct. Anal. 5 (2014), no. 1, 1–9
A nnals of F unctional A nalysis
ISSN: 2008-8752 (electronic)
URL:www.emis.de/journals/AFA/
HÖLDER TYPE INEQUALITIES ON HILBERT C ∗ -MODULES
AND ITS REVERSES
YUKI SEO
Dedicated to Professor Tsuyoshi Ando in celebration of his distinguished achievements in
Matrix Analysis and Operator Theory
Communicated by J. Chmieliński
Abstract. In this paper, we show Hilbert C ∗ -module versions of Hölder–
McCarthy inequality and its complementary inequality. As an application, we
obtain Hölder type inequalities and its reverses on a Hilbert C ∗ -module.
1. Introduction
The Hölder inequality is one of the most important inequalities in functional
analysis. If a = (a1 , . . . , an ) and b = (b1 , . . . , bn ) are n-tuples of nonnegative
numbers, and 1/p + 1/q = 1, then
!1/p
!1/q
n
n
n
X
X
X
p
q
ai b i ≤
ai
bi
for all p > 1
i=1
i=1
i=1
and
n
X
i=1
ai b i ≥
n
X
i=1
!1/p
api
n
X
!1/q
bqi
for all p < 0 or 0 < p < 1.
i=1
Non-commutative versions of the Hölder inequality and its reverses have been
studied by many authors. Ando [1] showed the Hadamard product version of a
Hölder type. Ando and Hiai [2] discussed the norm Hölder inequality and the
matrix Hölder inequality. Mond and Shisha [15], Fujii, Izumino, Nakamoto and
Date: Received: 25 March 2013; Accepted: 27 April 2013.
2010 Mathematics Subject Classification. Primary 46L08; Secondary 47A63, 47A64.
Key words and phrases. Hölder–McCarthy inequality, Hölder inequality, Hilbert C ∗ -modules,
geometric operator mean.
1
2
Y. SEO
Seo [7], and Izumino and Tominaga [11] considered the vector state version of a
Hölder type and its reverses. Bourin, Lee, Fujii and Seo [3] showed the geometric
operator mean version, and so on.
In this paper, as a generalization of the vector state version due to [7], we show
Hilbert C ∗ -module versions of Hölder–McCarthy inequality and its complementary inequality. As an application, we obtain Hölder type inequalities and its
reverses on a Hilbert C ∗ -module.
2. preliminary
Let B(H) be the C ∗ -algebra of all bounded linear operators on a Hilbert space
H, and A be a unital C ∗ -algebra of B(H) with the unit element e. For a ∈ A ,
1
we denote the absolute value of a by |a| = (a∗ a) 2 . For positive elements a, b ∈ A
and t ∈ [0, 1], the t-geometric mean of a and b in the sense of Kubo–Ando theory
[12] is defined by
1
1
1
1 t
a ]t b = a 2 a− 2 ba− 2 a 2
for a > 0, i.e., a is invertible. In the case of non-invertible, since a ]t b satisfies the
upper semicontinuity, we define a ]t b = limε→+0 (a + εe) ]t (b + εe) in the strong
operator topology. Hence a ]t b ∈ A 00 in general, where A 00 is the bi-commutant
of A . In the case of t = 1/2, we denote a ]1/2 b by a ] b simply. The operator
geometric mean has the symmetric property: a ]t b = b ]1−t a, and a ]t b = a1−t bt
for commuting a and b.
A complex linear space X is said to be an inner product A -module (or a preHilbert A -module) if X is a right A -module together with a C ∗ -valued map
(x, y) 7→ hx, yi : X × X → A such that
(i)
(ii)
(iii)
(iv)
hx, αy + βzi = αhx, yi + βhx, zi (x, y, z ∈ X , α, β ∈ C),
hx, yai = hx, yia (x, y ∈ X , a ∈ A ),
hy, xi = hx, yi∗ (x, y ∈ X ),
hx, xi ≥ 0 (x ∈ X ) and if hx, xi = 0, then x = 0.
The linear structures of A and X are assumed to be compatible. If X satisfies
all conditions for an inner-product A -module except for the second part of (iv),
then we call X a semi-inner product A -module.
∗
Let X be an inner product
p A -module over a unital C -algebra A . We define
the norm of X by k x k:= k hx, xi k for x ∈ X , where the latter norm denotes
the C ∗ -norm of A . If X is complete with respect to this norm, then X is called
a Hilbert A -module. An element x of the Hilbert A -module is called nonsingular
if the element hx, xi ∈ A is invertible. For more details on Hilbert C ∗ -modules,
see [13, 14].
In [6], from a viewpoint of operator geometric mean, we showed the following
new Cauchy–Schwarz inequality:
Theorem 2.1 (Cauchy–Schwarz inequality). Let X be a semi-inner product A module over a unital C ∗ –algebra A . If x, y ∈ X such that the inner product
hx, yi has a polar decomposition hx, yi = u|hx, yi| with a partial isometry u ∈ A ,
HÖLDER TYPE INEQUALITIES ON HILBERT C ∗ -MODULES
3
then
|hx, yi| ≤ u∗ hx, xiu ] hy, yi.
(2.1)
Under the assumption that X is an inner product A -module and y is nonsingular, the equality in (2.1) holds if and only if xu = yb for some b ∈ A .
Next we review the basic concepts of adjointable operators on a Hilbert C ∗ module. Let X be a Hilbert C ∗ -module over a unital C ∗ -algebra A . Let
EndA (X ) denote the set of all bounded C-linear A -homomorphism from X
to X . Let T ∈ EndA (X ). We say that T is adjointable if there exists a
T ∗ ∈ EndA (X ) such that hT x, yi = hx, T ∗ yi for all x, y ∈ X . Let L(X ) denote
the set of all adjointable operators from X to X . Moreover, we define its norm
by
1
k T k= sup{k hT x, T xi k 2 :k x k≤ 1}.
Then L(X ) is a C ∗ -algebra. The symbol I stands for the identity operator in
L(X ). The following lemma due to Paschke [16] is very important:
Lemma 2.2. Let X be a Hilbert C ∗ -module and let T be a bounded A -linear
operator on X . The following conditions are equivalent:
(1) T is a positive element of L(X );
(2) hx, T xi ≥ 0 for all x in X .
In [8], we showed the following generalized Cauchy–Schwarz inequality on a
Hilbert C ∗ -module by virtue of (2.1) and Lemma 2.2:
Theorem 2.3 (generalized Cauchy–Schwarz inequality). Let T be a positive
operator in L(X ). If x, y ∈ X such that hx, T yi has a polar decomposition
hx, T yi = u|hx, T yi| with a partial isometry u ∈ A , then
|hx, T yi| ≤ u∗ hx, T xiu ] hy, T yi.
(2.2)
Under the assumption that hy, T yi is invertible, the equality in (2.2) holds if and
1
1
only if T 2 (xu) = T 2 (yb) for some b ∈ A .
3. Hölder–McCarthy inequality
In this section, we show two Hilbert C ∗ -module versions of Hölder–McCarthy
inequality and its complementary inequality. For convenience, we use the notation
\t for the binary operation
1
1
1
1 t
a \t b = a 2 a− 2 ba− 2 a 2
for t 6∈ [0, 1],
whose formula is the same as ]t .
Theorem 3.1. Let T be a positive operator in L(X ) and x a nonsingular element
of X .
(1) If p ≥ 1, then hx, T xi ≤ hx, xi ]1/p hx, T p xi.
(2) If p ≤ −1 or 1/2 ≤ p ≤ 1, then hx, xi \1/p hx, T p xi ≤ hx, T xi.
4
Y. SEO
Proof. For a nonsingular element x of X , Put
1
1
Φx (X) = hxhx, xi− 2 , Xxhx, xi− 2 i
for X ∈ L(X ).
Then Φx is a unital positive linear map from L(X ) to A .
Suppose that p ≥ 1. Since t1/p is operator concave, it follows from [4, 5] that
Φx (T 1/p ) ≤ Φx (T )1/p and this implies
1
1
1
1 1/p
hx, xi− 2 hx, T 1/p xihx, xi− 2 ≤ hx, xi− 2 hx, T xihx, xi− 2
and
1
1
1 1/p
1
hx, T 1/p xi ≤ hx, xi 2 hx, xi− 2 hx, T xihx, xi− 2
hx, xi 2
(3.1)
= hx, xi ]1/p hx, T xi.
Replacing T by T p in (3.1), we have (1).
Suppose that p ≤ −1 or 1/2 ≤ p ≤ 1. Since −1 ≤ 1/p < 0 or 1 ≤ 1/p ≤ 2, we
1
1
have Φx (T ) p ≤ Φx (T p ) by the operator convexity of t1/p and this implies
p1
1
1
1
1
1
hx, xi− 2 hx, T xihx, xi− 2
≤ hx, xi− 2 hx, T p xihx, xi− 2 .
Hence it follows that
1
hx, xi \1/p hx, T xi ≤ hx, T p xi
and replacing T by T p in (3.2) we have (2).
(3.2)
Remark 3.2. The inequality (2) of Theorem 3.1 does not hold for 0 < p < 1/2
in general. In fact, we give a simple counterexample to the case of p = 1/3 as
follows: Put
X = M4 (C) = M2 (M2 (C)) and A = M2 (C) ⊕ M2 (C)
and
X Y
X 0
Φ(
)=
Z W
0 W
for X, Y, Z, W ∈ M2 (C). Then X is a Hilbert A -module with an inner product
hx, yi = Φ(x∗ y) for x, y ∈ X . Let




2 1 1 1
1/2 0 0 0
1 1 1 1
 0 2 0 0



z=
1 1 1 1 and x =  0 0 1 0 .
1 1 1 1
0 0 0 1
If T = Tz is defined by Tz y = zy for all y ∈ X , then T is a positive operator in
L(X ) and
13 8
4 4
−1/2
−1/2 3
hx, xi
hx, T xihx, xi
=
⊕
8 5
4 4
and
−1/2
hx, xi
3
−1/2
hx, T xihx, xi
=
29 22
17 17
⊕
,
22 17
17 17
HÖLDER TYPE INEQUALITIES ON HILBERT C ∗ -MODULES
5
so that
hx, xi−1/2 hx, T 3 xihx, xi−1/2 − hx, xi−1/2 hx, T xihx, xi−1/2
16 14
13 13
=
⊕
6≥ 0 ⊕ 0.
14 12
13 13
3
Next, we show a complementary part of Theorem 3.1. For this, we need the
generalized Kantorovich constant K(α, β, p) for 0 < α < β, which is defined by
p
αβ p − βαp
p − 1 β p − αp
(3.3)
K(α, β, p) =
(p − 1)(β − α)
p αβ p − βαp
for any real number p ∈ R, see also [10, Definition 2.2]. The constant K(α, β, p)
satisfies 0 < K(α, β, p) ≤ 1 for 0 ≤ p ≤ 1 and K(α, β, p) ≥ 1 for p 6∈ [0, 1]. For
more details on the generalized Kantorovich constant, see [10, Chapter 2.7].
Theorem 3.3. Let T be a positive invertible operator in L(X ) such that αI ≤
T ≤ βI for some scalars 0 < α < β, and x a nonsingular element of X .
(1) If p ≥ 1, then
hx, xi ]1/p hx, T p xi ≤ K(α, β, p)1/p hx, T xi.
(2) If p ≤ −1 or 1/2 ≤ p ≤ 1, then
hx, T xi ≤ K(αp , β p , 1/p)hx, xi \1/p hx, T p xi,
where the generalized Kantorovich constant K(α, β, p) is defined by (3.3).
1
1
Proof. For a nonsingular element x of X , put Φx (X) = hxhx, xi− 2 , Xxhx, xi− 2 i
for X ∈ L(X ). Then Φx : L(X ) 7→ A is a unital positive linear map.
Suppose that p ≥ 1. It follows from [10, Lemma 4.3] that
Φx (T p ) ≤ K(α, β, p)Φx (T )p .
This implies
hx, xi ]1/p hx, T p xi ≤ K(α, β, p)1/p hx, T xi
and we have (1).
In the case of p ≤ −1 or 1/2 ≤ p ≤ 1, since −1 ≤ 1/p < 0 or 1 ≤ 1/p ≤ 2,
it follows that Φx (T 1/p ) ≤ K(α, β, 1/p)Φx (T )1/p . Similarly we have the desired
inequality (2).
Next, we discuss Hölder–McCarthy type inequalities on a Hilbert C ∗ -module
outside intervals of Theorem 3.1.
Corollary 3.4. Let T be a positive invertible operator in L(X ) such that αI ≤
T ≤ βI for some scalars 0 < α < β, and x a nonsingular element of X . If
−1 < p < 0 or 0 < p < 1/2, then
K(αp , β p , 1/p)−1 hx, T xi ≤ hx, xi \1/p hx, T p xi ≤ K(αp , β p , 1/p)hx, T xi,
where the generalized Kantorovich constant K(α, β, p) is defined by (3.3).
6
Y. SEO
Proof. For a unital positive linear map Φx from L(X ) to A , it follows from [10,
Lemma 4.3] that for −1 < p < 0 or 0 < p < 1/2
K(α, β, 1/p)−1 Φx (T )1/p ≤ Φx (T 1/p ) ≤ K(α, β, 1/p)Φx (T )1/p .
Hence we have this theorem as in the proof of Theorem 3.3.
Similarly we have the following Hölder–McCarthy type inequality on a Hilbert
C ∗ -module and its complementary inequality as follows:
Theorem 3.5. Let T be a positive invertible operator in L(X ) such that αI ≤
T ≤ βI for some scalars 0 < α < β. Then for 0 < p < 1
K(α, β, p)hx, xi ]p hx, T xi ≤ hx, T p xi ≤ hx, xi ]p hx, T xi
for every nonsingular element x ∈ X , where K(α, β, p) is defined by (3.3).
4. Hölder inequality
As an application of Theorem 3.1 and Theorem 3.3, we show Hölder type
inequalities on a Hilbert C ∗ -module and its reverses.
Theorem 4.1. Let A and B be positive invertible operators in L(X ) and x a
nonsingular element of X , and p1 + 1q = 1.
(1) If p > 1, then
hx, B q ]1/p Ap xi ≤ hx, B q xi ]1/p hx, Ap xi
(4.1)
hx, Ap ]1/q B q xi ≤ hx, Ap xi ]1/q hx, B q xi.
(2) If p ≤ −1 or 12 ≤ p < 1, then
(4.2)
hx, B q \1/p Ap xi ≥ hx, B q xi \1/p hx, Ap xi
(4.3)
hx, Ap \1/q B q xi ≥ hx, Ap xi \1/q hx, B q xi.
(4.4)
or
or
q
q
q
1
Proof. Replacing x and T by B 2 x and (B − 2 Ap B − 2 ) p in (1) of Theorem 3.1
respectively, we have (4.1) of Theorem 4.1. By (4.1) and the symmetric property
of t-geometric mean, we have (4.2). The latter (4.3) and (4.4) are proved similarly.
By Theorem 3.5, we have the following weighted version of Cauchy type inequality on a Hilbert C ∗ -module.
Theorem 4.2. Let A and B be positive invertible operators in L(X ) such that
αI ≤ A, B ≤ βI for some scalars 0 < α < β. Then for 0 < p < 1
α2 β 2
, , p)hx, B 2 xi ]p hx, A2 xi ≤ hx, A2 ]p B 2 xi ≤ hx, B 2 xi ]p hx, A2 xi
β 2 α2
for every nonsingular element x ∈ X .
K(
Proof. Replace x and T by Bx and B −1 A2 B −1 in Theorem 3.5 respectively. Since
2
α2
I ≤ B −1 A2 B −1 ≤ αβ 2 , the theorem follows.
β2
HÖLDER TYPE INEQUALITIES ON HILBERT C ∗ -MODULES
7
If we put p = 1/2 in Theorem 4.2, then we have the following Pólya-Szegö type
inequality on a Hilbert C ∗ -module which is regarded as a reverse of Cauchy type
inequality, also see [8, Theorem 3.3].
Corollary 4.3. Let A and B be positive invertible operators in L(X ) such that
αI ≤ A, B ≤ βI for some scalars 0 < α < β. Then
α+β
hx, Axi ] hx, Bxi ≤ √ hx, A ] Bxi
2 αβ
for every nonsingular element x ∈ X .
Next, we show a complementary version of Theorem 4.1.
Theorem 4.4. Let A and B be positive invertible operators in L(X ) such that
αI ≤ A, B ≤ βI for some scalars 0 < α < β, and x a nonsingular element of X
and p1 + 1q = 1.
(1) If p > 1, then
q
p
hx, B xi ]1/p hx, A xi ≤ K
α
β q−1
,
p1
β
αq−1
,p
hx, B q ]1/p Ap xi.
(2) If p ≤ −1 or 1/2 ≤ p < 1, then
p p −1
α β 1
q
p
hx, B xi \1/p hx, A xi ≥ K
, ,
hx, B q \1/p Ap xi.
β q αq p
q
q
q
1
Proof. Replace x and T by B 2 x and (B − 2 Ap B − 2 ) p in (1) of Theorem 3.3 respecq
q 1
tively. Since α/β q−1 I ≤ (B − 2 Ap B − 2 ) p ≤ β/αq−1 I, we have (1) of Theorem 4.4.
The latter (2) are proved similarly.
Next, we discuss Hölder type inequalities in a complementary interval of Theorem 4.1.
Corollary 4.5. Let A and B be positive invertible operators in L(X ) such that
αI ≤ A, B ≤ βI for some scalars 0 < α < β, and x a nonsingular element of X
and p1 + 1q = 1. If −1 < p < 0 or 0 < p < 12 , then
p p −1
α β 1
K
, ,
hx, B q \1/p Ap xi ≤ hx, B q xi \1/p hx, Ap xi
β q αq p
p p α β 1
≤K
, ,
hx, B q \1/p Ap xi.
β q αq p
q
q
q 1
Proof. Replacing x and T by B 2 x and B − 2 Ap B − 2 p in Corollary 3.4 respectively, we have this theorem.
5. Weighted Cauchy–Schwarz inequality
In this section, we discuss weighted Cauchy–Schwarz inequality on a Hilbert
C -module. We cite [9] for the case of the Hilbert space operator.
For T ∈ L(X ), we denote the range of T and the kernel of T by R(T ) and
N (T ), respectively. A closed submodule M of X is said to be complemented if
∗
8
Y. SEO
X = M ⊕ M ⊥ . Suppose that the closures of the ranges of T and T ∗ are both
complemented. Then it follows from [13, page 30] that T has a polar decomposition T = U |T | with a partial isometry U ∈ L(X ) and N (U ) = N (|T |). Also, we
showed in [8, Lemma 6.1] that
|T ∗ |q = U |T |q U ∗
for any positive number q.
(5.1)
As a generalization of Theorem 2.3, we have the following inequality.
Theorem 5.1 (Weighted Cauchy–Schwarz Inequality). Let T be an operator in
L(X ) such that the closures of the ranges of T and T ∗ are both complemented.
If x, y ∈ X such that hT x, yi has a polar decomposition hT x, yi = u|hT x, yi| with
a partial isometry u ∈ A , then the following inequality holds
|hT x, yi| ≤ u∗ hx, |T |2α xiu ] hy, |T ∗ |2β yi
(5.2)
for any α, β ∈ [0, 1] and α + β = 1. In particular,
|hT x, yi| ≤ u∗ hx, |T |2 xiu ] hy, U U ∗ yi
and
|hT x, yi| ≤ u∗ hx, U ∗ U xiu ] hy, |T ∗ |2 yi.
Moreover, under the assumption that hy, |T ∗ |2β yi is invertible for β ∈ [0, 1], the
equality in (5.2) holds if and only if T xu = |T ∗ |2β yb for some b ∈ A .
Proof. In the case of α = 0 or 1, it follows from Theorem 2.1 that
|hT x, yi| = |h|T |x, U ∗ yi| ≤ u∗ hx, |T |2 xiu ] hy, U U ∗ yi
and
|hT x, yi| = |hx, |T |U ∗ yi| = |hx, U ∗ U |T |U ∗ yi| = |hU x, |T ∗ |yi|
≤ u∗ hU x, U xiu ] h|T ∗ |y, |T ∗ |yi = u∗ hx, U ∗ U xiu ] hy, |T ∗ |2 yi
by (5.1).
In the case of 0 < α < 1, we have
|hT x, yi| = |hU |T |x, yi| = |h|T |α x, |T |β U ∗ yi| by α + β = 1
≤ u∗ hx, |T |2α xiu ] hy, U |T |2β U ∗ yi by Theorem 2.1
= u∗ hx, |T |2α xiu ] hy, |T ∗ |2β yi. by (5.1).
Next, we consider the equality conditions in (5.2). Since hT x, yi = h|T |α x, |T |β U ∗ yi
and hy, |T ∗ |2β yi is invertible for β ∈ [0, 1], it follows from Theorem 2.1 that
the equality in (5.2) holds if and only if |T |α xu = |T |β U ∗ yb for some b ∈ A .
Since |T |x = 0 if and only if |T |1/2 x = 0, it follows that N (|T |) = N (|T |q )
for any positive real numbers
q > 0. If |T |β |T |α xu − |T |β U ∗ yb = 0, then
|T |q |T |α xu − |T |β U ∗ yb = |T |α+q xu − |T |β+q U ∗ yb = 0 for any q > 0 and this
implies |T |α xu − |T |β U ∗ yb = 0. Therefore we have the following implications:
|T |α xu = |T |β U ∗ yb ⇐⇒ |T |α+β xu = |T |2β U ∗ yb ⇐⇒ U |T |xu = U |T |2β U ∗ yb
⇐⇒ T xu = |T ∗ |2β yb
by (5.1).
HÖLDER TYPE INEQUALITIES ON HILBERT C ∗ -MODULES
If we put α = β =
1
2
9
in Theorem 5.1, then we have the following inequality.
Theorem 5.2. Let T be an operator in L(X ) such that the closures of the ranges
of T and T ∗ are both complemented. If x, y ∈ X such that hT x, yi has a polar
decomposition hT x, yi = u|hT x, yi| with a partial isometry u ∈ A , then
|hT x, yi| ≤ u∗ hx, |T |xiu ] hy, |T ∗ |yi.
(5.3)
∗
Moreover, under the assumption that hy, |T |yi is invertible, the equality in (5.3)
holds if and only if T xu = |T ∗ |yb for some b ∈ A .
Acknowledgement. The authors would like to express their cordial thanks
to the referee for his/her valuable suggestions.
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Department of Mathematics Education, Osaka Kyoiku University, 4-698-1 Asahigaoka, Kashiwara, Osaka 582-8582 Japan.
E-mail address: yukis@cc.osaka-kyoiku.ac.jp